"... The work reported here lays the foundations of data exchange in the presence of probabilistic data. This requires rethinking the very basic concepts of traditional data exchange, such as solution, universal solution, and the certain answers of target queries. We develop a framework for data exchange ..."

The work reported here lays the foundations of data exchange in the presence of probabilistic data. This requires rethinking the very basic concepts of traditional data exchange, such as solution, universal solution, and the certain answers of target queries. We develop a framework for data exchange over probabilistic databases, and make a case for its coherence and robustness. This framework applies to arbitrary schema mappings, and finite or countably infinite probability spaces on the source and target instances. After establishing this framework and formulating the key concepts, we study the application of the framework to a concrete and practical setting where probabilistic databases are compactly encoded by means of annotations formulated over random Boolean variables. In this setting, we study the problems of testing for the existence of solutions and universal solutions, materializing such solutions, and evaluating target queries (for unions of conjunctive queries) in both the exact sense and the approximate sense. For each of the problems, we carry out a complexity analysis based on properties of the annotation, in various classes of dependencies. Finally, we show that the framework and results easily and completely generalize to allow not only the data, but also the schema mapping itself to be probabilistic.

.... . . , bki). DEFINITION 3.3. LetK be a class of algebras. The embedding problem for K asks: given a finite partial algebra B, is B embeddable in some algebra A in K? In a series of papers, including =-=[8, 9, 10]-=-, Evans investigated the embedding problem for varieties, that is, classes of algebras axiomatized by identities. One of his main findings is that the word problem for a variety K is decidable if and ...

"... In this paper we compare three approaches to polynomial time decidability for uniform word problems for quasivarieties. Two of the approaches, by Evans and Burris, respectively, are semantical, referring to certain embeddability and axiomatizability properties. The third approach is more proof-theor ..."

In this paper we compare three approaches to polynomial time decidability for uniform word problems for quasivarieties. Two of the approaches, by Evans and Burris, respectively, are semantical, referring to certain embeddability and axiomatizability properties. The third approach is more proof-theoretic in nature, inspired by McAllester&apos;s concept of local inference. We define two closely related notions of locality for equational Horn theories and show that both the criteria by Evans and Burris lie in between these two concepts. In particular, the variant we call stable locality will be shown to subsume both Evans&apos; and Burris&apos; method.

...s, locality and Evans' criterion coincide. We also show that the weaker form of Evans' criterion with satisfaction replaced by weak satisfaction is equivalent with locality. Looking at the proofs in (=-=Evans 1951-=-) it is not surprising that some sort of relation exists between embeddability and locality. However the precise details are not so straightforward, the reason being that Evan's notion of validity, in...

"... The paper presents a modular superposition calculus for the combination of firstorder theories involving both total and partial functions. The modularity of the calculus is a consequence of the fact that all the inferences are pure – only involving clauses over the alphabet of either one, but not bo ..."

The paper presents a modular superposition calculus for the combination of firstorder theories involving both total and partial functions. The modularity of the calculus is a consequence of the fact that all the inferences are pure – only involving clauses over the alphabet of either one, but not both, of the theories – when refuting goals represented by sets of pure formulae. The calculus is shown to be complete provided that functions that are not in the intersection of the component signatures are declared as partial. This result also means that if the unsatisfiability of a goal modulo the combined theory does not depend on the totality of the functions in the extensions, the inconsistency will be effectively found. Moreover, we consider a constraint superposition calculus for the case of hierarchical theories and show that it has a related modularity property. Finally we identify cases where the partial models can always be made total so that modular superposition is also complete with respect to the standard (total function) semantics of the theories. 1

...idity We consider extensions of a base theory with partial functions. The semantics for partial functions we consider is known as “Evans validity”. It was introduced, in the equational case, by Evans =-=[10,11]-=-, while identifying situations when the uniform word problem in classes of algebras axiomatized by a set E of identities is decidable in PTIME. We briefly present Evans’ method and his motivation for ...

...model of the axioms in K can be extended to a total algebra model of K, then the uniform word problem for K is decidable in polynomial time (this generalizes previous results by Skolem [21] and Evans =-=[6]-=-). A link between these embeddability properties and a proof theoretic concept (local theories) was established by Ganzinger [8]. In [25], we extended these results to special types of theory extensio...

"... Abstract. We present an overview of results on hierarchical and modular reasoning in complex theories. We show that for a special type of extensions of a base theory, which we call local, hierarchic reasoning is possible (i.e. proof tasks in the extension can be hierarchically reduced to proof tasks ..."

Abstract. We present an overview of results on hierarchical and modular reasoning in complex theories. We show that for a special type of extensions of a base theory, which we call local, hierarchic reasoning is possible (i.e. proof tasks in the extension can be hierarchically reduced to proof tasks w.r.t. the base theory). Many theories important for computer science or mathematics fall into this class (typical examples are theories of data structures, theories of free or monotone functions, but also functions occurring in mathematical analysis). In fact, it is often necessary to consider complex extensions, in which various types of functions or data structures need to be taken into account at the same time. We show how such local theory extensions can be identified and under which conditions locality is preserved when combining theories, and we investigate possibilities of efficient modular reasoning in such theory combinations. We present several examples of application domains where local theories and local theory extensions occur in a natural way. We show, in particular, that various phenomena analyzed in the verification literature can be explained in a unified way using the notion of locality. 1

... that every partially-ordered set (where ∨ and ∧ are partially defined) embeds into a lattice. A similar idea was used by Evans in the study of classes of algebras with a PTIME decidable word problem =-=[12]-=-. The idea was extended by Burris [8] to quasi-varieties of algebras. He proved that if a quasi-variety axiomatized by a set K of Horn clauses has the property that every finite partial algebra which ...

...bra A is residually finite if for any x # y in A, there is a homomorphism a of A onto a finite algebra such that xct # yu. For the notion of an incomplete or partial algebra in a variety, we refer to =-=[4, 6]-=-. We say that an algebra A in a variety V has the finite embeddability property if any finite incomplete F-algebra contained in A is embeddable in a finite F-algebra. A variety V is said to have the f...

"... A canonical form for elements of a lattice freely generated by a partial lattice is given. This form agrees with Whitman's canonical form for free lattices when the partial lattice is an antichain. The connection between this canonical form and the arithmetic of the lattice is given. For exa ..."

A canonical form for elements of a lattice freely generated by a partial lattice is given. This form agrees with Whitman&apos;s canonical form for free lattices when the partial lattice is an antichain. The connection between this canonical form and the arithmetic of the lattice is given. For example, it is shown that every element of a finitely presented lattice has only finitely many minimal join representations and that every join representation can be refined to one of these. An algorithm is given which decides if a given element of a finitely presented lattice has a cover and finds them if it does. An example is given of a nontrivial, finitely presented lattice with no cover at all.

"... If C is a class of groups closed under taking subgroups, we show that the decidability of the uniform word problem for C is implied by the decidability of the membership problem for the class of nite Rees quotients of E-unitary inverse semigroups with maximal group image in C. The converse is sh ..."

If C is a class of groups closed under taking subgroups, we show that the decidability of the uniform word problem for C is implied by the decidability of the membership problem for the class of nite Rees quotients of E-unitary inverse semigroups with maximal group image in C. The converse is shown if C is a pseudovariety. When C is a pseudovariety, the above problems are shown to be equivalent to the problem of embedding a nite labeled graph in the Cayley graph of a group in C. This latter problem is shown to be equivalent to deciding whether a nite labeled graph is a Schutzenberger graph of an E-unitary inverse semigroup with maximal group image in C. 1.

...thers. It is known [6] that the uniform word problem for a pseudovariety of groups C is equivalent to the decidability of its universal theory. A result of THE UNIFORM WORD PROBLEM FOR GROUPS 3 Evans =-=[4]-=- shows that it is also equivalent to the decidability of the problem of embedding a partial group into a group in C. In Theorem 1.1, if C is just assumed to be closed under S then (2) and (3) are equi...